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Let $$\matrix{ A& \mathop{\longrightarrow}\limits^f &B\\ \Big\downarrow & & \Big\downarrow\\ C&\mathop{\longrightarrow}\limits_g &D }$$ Be a pushout diagram in a category $\mathcal C$. If $f$ is monic, is $g$ also monic?

I have known that this holds in an abelian category. Is it true for a general category? If so, how to prove it?

If if fails, could anyone give me a counterexample? And what conditions should we impose on the category $\mathcal C$ to ensure that this is true?


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Monics that are preserved under all pushouts are called (in French) monomorphismes universels (the dual concept, épimorphismes universels, is introduced in SGA 4 (I.10.3) along with an extensive list of other properties epimorphisms may or may not have). – t.b. Jun 6 '12 at 8:14
@t.b.: 1+. And there is a (rather sparse) nlab article on that concept: – Martin Brandenburg Jun 6 '12 at 8:18
@Martin: Thanks! I wasn't sure whether that name caught on or not and last time I checked that nlab page didn't exist... Obviously some people still use it :) (+1 for your nice answer, by the way!) – t.b. Jun 6 '12 at 8:31

In the category of commutative rings, this pushout diagram means that $D = C \otimes_A B$, and you ask: If $A \to B$ is monic (which is the case iff the underlying map is injective), is the same true for the cobase change $C \to C \otimes_A B$ mapping $c \mapsto c \otimes 1$? Well this is true when $A \to C$ is flat, but in general its terribly false: If $C=A/I$ for some ideal $I \subseteq A$, then this is the case iff $IB \cap A = I$. And this is rather rare. Take for instance $A=\mathbb{Z}$, $B=\mathbb{Q}$, and $I$ any non-trivial ideal.

I don't know if there are any reasonable and non-trivial assumptions on a category which makes the statement true. In the category of sets the statement is true (and probably also in every topos). Hence, it is also true in many other concrete categories whose forgetful functor preserves pushouts and monics, for instance the category of topological spaces. In fact, this property appears in the realm of Waldhausen categories. There you require that the class of cofibrations is stable under cobase change. Often one imagines these cofibrations as "nice" monomorphisms.

The statement is also true in every dual of an algebraic category with the property that epis coincide with surjective homomorphisms, since surjective homomorphisms are stable under base change: If $B \to A$ is surjective, then for every map $C \to A$ it is clear that also $C \times_A B \to C$ is surjective. Is it true in the dual category of the category of commutative rings (epimorphisms of rings are rather complicated, see here), i.e. the category of affine schemes?

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It is indeed true in any topos, because toposes are adhesive. – Zhen Lin Jun 6 '12 at 8:33
You are absolutely right! Thanks so much for your insightful and detailed answer! – Yang Zhou Jun 6 '12 at 8:56

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